Higher-Order Weak Formulation for Vibration Analysis of Anisotropic Doubly-Curved Shells with Holes and Discontinuities

This work focuses on the application of two-dimensional higher-order theories within the Equivalent Single Layer (ESL) approach for the free vibration analysis of anisotropic doubly-curved shells with irregular bi-connected domains, holes, and cracks. To this end, the dynamic governing equations are derived from the Hamilton principle in curvilinear principal coordinates, using a higher-order Lagrange interpolation of the unknown variables. An isogeometric mapping technique is employed to account for arbitrarily-shaped structures. A constitutive relation valid for generally anisotropic materials is considered within the theory, and a generalized kinematic model, based on power and zigzag functions, is adopted along the thickness direction. A numerical solution is derived with the Generalized Differential Quadrature (GDQ) method. Furthermore, a finite element implementation of the theory is carried out using higher-order Lagrange and Hermite shape functions to interpolate the solution within rectangular and triangular elements which discretize the parametric domain. The model is validated against some reference solutions obtained from finite elements, and the convergence of the results from numerical method is assessed. The accuracy of the solution is investigated for structures described by bi-connected domains, including panels with one or more holes of arbitrary shape. Furthermore, doubly-curved laminated shells are studied with internal line cracks. The procedure presented in this paper represents a significative advancement in the analysis of shell structures, since it provides a comprehensive finite element formulation for anisotropic doubly-curved shells, based on unified formulations, which can be used even for curved laminated structures characterized by holes, discontinuities, cracks, and arbitrary shape.